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We prove that the derivative map $d colon mathrm{Diff}_partial(D^k) to Omega^kSO_k$, defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for $k = 11$ we prove that the following homomorphism is non-zero: $$ d_* colon pi_5mathrm{Diff}_partial(D^{11}) to pi_{5}Omega^{11}SO_{11} cong pi_{16}SO_{11} $$ As a consequence we give a counter-example to a conjecture of Burghelea and Lashof and so give an example of a non-trivial vector bundle $E$ over a sphere which is trivial as a topological $mathbb{R}^k$-bundle (the rank of $E$ is $k=11$ and the base sphere is $S^{17}$.) The proof relies on a recent result of Burklund and Senger which determines those homotopy 17-spheres bounding $8$-connected manifolds, the plumbing approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an explicit construction of low-codimension embeddings of certain homotopy spheres.
The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enou
A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N --> N where N is a manifold. The action of this operad on EC(j,M) (self embeddings R^j x M --> R^j x M with support i
We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation no
The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.
For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contract